Finally, we can find the last digit of pi
This also means I can finally crack Jeff Deanβs password.
A formula for the nth digit of π and π^n
71β80 of 143 posts
Re: A formula for the nth digit of π and π^n
#72But... pi is in the formula. If you already know it, then why bother?
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Re: A formula for the nth digit of π and π^n
#73Finally, we can find the last digit of pi
The last digit of pi is e.
Re: A formula for the nth digit of π and π^n
#74It is all curious, but it seems that procedure that calculates the n-th digit using some other functions that require O(n) calculations (i.e. Bernoulli numbers) is not that exciting, as it's just a speedup comparing to a naive calculation (maybe a big one but still). Although because pi is a transcendental number maybe it is impossible to have an algorithm to return the n-th digit in O(1) operations? Does anyone knowβ¦
Even rational numbers which have repeating digits at the end are not O(1) because in order to calculate the n-th digit you need to calculate n modulo k where k is the length until the number repeats its digits, which needs to read the entire input number n (thus runs in O(log(n)).
The only exception are rational numbers with k being a power of two.
Take 0.1010101010β¦, you can get the n-th digit by reading off the smallest digit of n and checking whether it is a 0 or a 1.
There are transcendental numbers where the n-th digit can be computed in O(log n) like Champerowne constant though.
Re: A formula for the nth digit of π and π^n
#75Earlier quoted context omitted.
Spoiler: it's zero, recurring.
Can you prove it's not nine, recurring?
Yes... proving that would require an infinite-capacity rounding mechanism, which cannot exist, because when trying to build it, you always run out of universe. Thusly, "impossible" is equivalent to "false", by default.
Re: A formula for the nth digit of π and π^n
#76Earlier quoted context omitted.
The article title says "powers of π", so presumably an arbitrary digit.
Iβm not sure thatβs right. The abstract and text say the nth digit of \pi^n, and the worked example on page 2 uses the same n=1000 to find the thousandth digit of \pi^{1000}.
IOW, the arXiv title might have lost the "nth" because of some formatting thing. The title inside the PDF says "A formula for nth digit of \pi and \pi^n"
Re: A formula for the nth digit of π and π^n
#77But... pi is in the formula. If you already know it, then why bother?
Not sure where you're seeing that. Did you misread a formula?
Re: A formula for the nth digit of π and π^n
#78But... pi is in the formula. If you already know it, then why bother?
It isn't.
Re: A formula for the nth digit of π and π^n
#79Earlier quoted context omitted.
I don't think that would be true seeing 1/3 = 0.3333.. , or am I reading this wrong?
3 recurring represents a particular quantity continuing forever. 0 recurring represents the end of a quantity, and the absence of any further quantity, forever. Eg: 0.012500000000000... The significant portion is 0.0125 - the recurring zeroes serve a mathematical role akin to that of a full-stop in a sentence. Hence zero being (jokingly, but in a sense truthfully) always the "last digit".
This is just an artifact of representing it in base-10. In base-3 0.0125 has the same value but would have a non-terminating representation of 0.00010001...
I will grant that in base-Ο, Ο is 10, however.
Re: A formula for the nth digit of π and π^n
#80Earlier quoted context omitted.
Οfs: Never worry about data again! - https://github.com/philipl/pifs
It's actually not known if pi contains every combination of digits (in any base 2 or greater) or not. It feels likely but really all we know is it's transcendental and seems pretty random from the parts we've generated. The library of babel could be a good "useless" backing though https://libraryofbabel.info/bookmark.cgi?hnexample
You just now convinced me how that site wasn't a sham. Thank you.
It takes the input, converts it into N, then calculates position using the N.
Thank you. I argued for a long time simply by being ignorant.
good bless ya sir.